Good idea coloring the flat and the rounded part in different colors for the black and white D10.
For the numbers, I use a gel ink pen to ink them. The gel stay in the numbers without expanding out of the indentation. The problem is finding a pen thin enough to reach the inside of the numbers...

For the D32, I love the vivid colors of Uni Posca. But I am curious to know which kind of numbering you are using (0-1-2-3, 1-2-3-5?)...

The football dice are numbered in such a way that they should produce realistic scores. There is also a chart you have to refer to determine which dice are used for a certain match. I can PM you with the details if you like. I made up a computer version of it some years ago which I'll dig out and send you although it was based on 14 sided dice and not 32.

Just for reference, here is the logic behind the numbering of the D18 Sphere:

The faces can be viewed as following 3 large circles (here in yellow, pink and blue):
The sum of the 8 numbers lying on any large circle is 76 (that is 8 times the average value, which is 9.5). There is nothing extraordinary here, this is only due to the fact that numbers on opposite faces sum to 19 (twice the average).
The most tricky property is that the sum of the 6 numbers surrounding any of the triangular rounded areas sum to 57 (six times 9.5).
There are 8 triangular areas that are in white in the scheme (including the white zone surrounding the 3 large circles). Each traingluar are is surrounded by 6 numbers: 3 on the edges (yellow, pink and blue) and 3 on the vertices (orange, purple and green). You can check: if you make the sum of any of such 6 numbers you'll find 57.

I guess there are several ways of numbering the D18 while complying with these properties.
But perhaps you can find another numbering with even more proprerties. Actually I tried also to have a nice symmetric numbering if you take the rest of the division by 3 of those numbers.

Let me know if you find something better! (but nothing to win this time )

Finally, the D32 has been numbered. It took a long time (it was designed in November 2010).

I decided this time to ignore the usual rule "numbers on opposite faces must sum to 33" and prefered the rule "difference between numbers on opposite faces must be 16". This allowed me to add interesting properties. In this case, you have faces surrounded by 6 faces and faces surrounded by 5 faces: if you take the number on a face surrounded by 5 faces and add to it the numbers on the 5 neighbours you will always got 99. This guarantees a certain fairness in the repartition of the numbers (basically, there is not an area of the die with more "value" than another).
As for the D33, there is a Regular Edition (diameter is 30mm, thickness is 1.5mm) and a Frosted Edition (diameter is 20 mm and thickness is 1.0 mm). Both version contain the support material.

Thanks Orangery.
For your reference, here is the layout of the D32 numbering:
There is just half of the die since the numbers on opposite faces can be deduced easily (the difference between numbers on opposite faces is 16).
For instance, around the number 24 that is on a pentagonal face, you have 5 numbers on hexagonal faces: 6, 29, 32 and 3 (that are visible) and the number on the opposite face of the hexagonal face having the number 21, that is 21-16=5.
And you can check that 24+6+29+32+3+5=99.

Still work in progress: the D22 Sphere.
It is constructed by intersecting a sphere with an unusual polyhedron. This polyhedron is based on a tetrahedron (and as the same symmetries as it): there are 3 faces for each vertex, one face by edge as well as the 4 original faces (3x4 + 6 + 4 = 22).

I was looking for a nice design for a D7. Generally the Pentagonal Prism is used for this number of faces. Unfortunately, I don't like the Truncated Sphere that results from this polyhedron. So I was looking for something different.
For a long time, I though no other interesting 7-sided polyhedron did exist, but after a lot of research with pen and paper and some calculations on Excel, I found out a polyhedron that leads to this Truncated Sphere D7:

I am more than happy with it. I think it has unexpected "partial" symmetries. It is difficult to explain, but as soon as I get a printed version, I will try to show you.

Well, the shape before being intersected with a sphere look like a rhombus for the upper face, 4 identical quadrangles as lateral faces and two triangles for the lower faces (1+4+2 = 7).
I don't know if it's new, but I know I have never seen it before...
In my opinion, it is less interesting that the corresponding truncated sphere, though...
And I think i should be able to create a new D7 with a 3-fold symmetry (based on the sum 1+3+3 instead of 1+4+2).
Stay tuned!

Here is a new addition to the Truncated Sphere dice family:
The D13 Sphere.
It has a 4-fold symmetry and is based on a 1+4+4+4 repartition of the faces (there is 1 single face surrounded by 4 faces at the bottom and 2 additional layers of 4 faces).
I had search along time how to make a D13 and there was no obvious solution: having this single face at one end allows new combinations, so expect more dice with odd number of faces to come...

The Alternative D24 is another D24 Sphere Dice, based on the Pentagonal Icositetrahedra.
This shape is more efficient than the classical D24 Sphere: for a same sphere radius, the faces are bigger. But the numbers are not located on faces (unless you read them on the bottom face, that is under the die).
The shape is chiral, so it comes in two flavours: right-handed and left-handed.
It's good to have choice...

The first one (in red) this the one with a 2-fold symmetry (1 rhombus + 4 quads + 2 isosceles triangles) and the second (in yellow) the one with a 3 fols symmetry (1 equilateral triangle + 3 irregular pentagons + 3 quads)

The first one was realy designed on paper before doing the calculations and trying it on the modeling software. I was trying to put the 7 circles on a sphere in an optimal way.
The second one is more classical. You can compare it to the D11 (5 layers: 1+3+3+3+1) but with less layers and with the layers not being symmetrical (there is no ending 1): 1+3+3.
I did not try it before because the equations are difficult to write and most of all difficult to solve. So I had to do a program to find approximations.
But the efforts are rewarding once you get a nice shape...

By the way both models can be seen as a tranformation of a cube:
- the first as a cube where one of the square face is cut into two in diagonal
- the second as a cube with one vertex cut to get an extra triangular face
Of course, this is just an initial construction: the vertices have to be moved to maximize the radii of the circles than can hold in the faces.

I used the same program I wrote for making the D13 to make the D19 Sphere:
Once again, the shape is inspired by the repulsive force polyhedra but is slightly different in order to maximize the diameter of the faces.
As for the D13 there is a face in one end that is "alone". The repartition of the faces by layers is 1+4+2+4+2+2+4.
There is only a 2-fold symmetry, but the underlying polyhedron is still attractive (even though its repulsive origin... ).

I am currently working on two D21.
The red one has a 2-fold symmetry, the yellow one a 4-fold symmetry.
You can see the underlying polyhedra (wireframe and solid).
The interesting point is that, although different, these two arrangements of 21 circles around the sphere seem as efficient (the diameter of the circles is the same for a given sphere). This is quite unsual...
The repartition of the faces in layers of faces with the same angle is
- 2-fold: 2+4+2+4+2+2+4+1
- 4-fold: 4+4+4+4+4+1

Before finishing the D21 (I still have to add the numbers), I wanted to make an alternative D6 Sphere.
Instead of being based on a cube, it is based on a double pyramid.
As a consequence, the numbers are not positionned on the faces but rather on the edges.

Yes, I think it is fair. It made of two pyramids with an equilateral triangular base stuck together. In this case, the height of the pyramid is caluclated to maximize the size of the circular faces obtained after the intersection with the sphere, but whatever the height, the double-pyramid is fair, so we can suppose that the truncated sphere is fair either. Of course this is true as long as the die does not stop on the rounded part...

For the 5-sided die, it is possible but it will not be a really surprising die: a triangular prism (once again with the appropriate height) intersecting a sphere: I looked for other shapes, without success.
I will make it very soon, I will finish the ones that are still unfinished (the two D21, the D22, and the D50) and I will probably make a pause with the dice in general and the truncated spheres in particular...

I have been following this blog on truncated spheres with great interest!

I like the idea of making dice out of truncated spheres. I would like to have dice that are „fair" or at least „optimum" in some sense.

In the strict sense, only isohedral dice are fair (platonic, catalan, dipyramids, trapeozohedra and some strange cousins, http://www.aleakybos.ch/sha.htm ). If a sphere is intersected with an isohedral polyhedron, I would call it fair, neglecting the rare case when such a die would stop on a curved part of the surface.

I got in touch with Bob who wrote the original visualization applet for repulsion force polyhedra and got his permission to modify it. I included various optimization criteria, the results of my endeavor are visible here: http://www.aleakybos.ch/sph_codes.htm

At this time the computation of the dual for some larger odd numbers is still buggy, but I will work on it

Actually, I know your website and I wanted to contact you because you mention that some shapes (like the Pentagonal Icositetrahedron) are not usable as die because basically the opposite of one face is not one face: this did not prevent me from positioning numbers where needed on the truncated spheres including out of the faces

I know the program from Bob (I did not contact him instead I was in touch with Martin Trump). Basicaly Repulsive Force polyhedra are great for inspiration, but I had to recalculate all the values probably because as you mentionned they use repulsion forces and I need packing. Generally the shapes are very close, only the values of the angles and sometime the shape of the faces are different (like irregular hexagons instead of triangles).

That's why I am very happy you wrote this new program.
On my side, the biggest difference I noted between the two methods was for the D13.
With Repulsion Force Polyedra, the D13 looks like a dodecahdron with an additional face replacing an edge (and repulsing all the other faces). This additional face is a rectangle, and the polyhedron has a 2 fold symmetry (identical when rotating it at 180°).
When I tried on my side to maximize the size of the face, this additional face turned square and the whole polyhedron had a 4 fold symmetry (identical when rotating it at 90°).
That's not what I obtain with your program... Perhaps I made a mistake in the design of my d13, when forcing the constraints.
What do you think?

Regards,

Magic
PS: is Alea your real first name? Alea sounds like aléa to me, the root of "aléatoire" meaning "random"...

Oops! Sorry with the "packing" option and 13 points, I obtain in effect the same shape as my D13. But it is indeed very different from the one obtained by minimizing energy and coverage.
With 7 points, the result with "packing" is similar to my D7 with 3-fold symmetry. It is interesting to note that I think I got the same diameter for the face (that is the separation angle between the two nearest adjacent circles) with the D7 with a 2-fold symmetry...
I like a lot the shape obtained with 8 point minimizing covering. I should be able to make a new die from this one.

And the D10 obtained by packing is different from any of the ones I have.

It seems that there are a lot of new shapes to explore...
Thank you for that!

I like the way you put numbers on dice whose faces are not parallel, like the D24 you mentioned. Could you use the same approach for numbering the isohedral D24 Pentagonal Icositetrahedron and the D60 Pentagonal Hexecontahedron (I mean the regular shapes, not the truncated spheres), That would be unique.

Comparing the various optimization criteria with my applet, I observed that the optimum solutions are identical for 4,5,6 points.

For 7 points you get a 1+3+3 shape with 3-fold symmetry with packing, and a 5-prism with all the others.

For 8 points packing and energy yield a 4-trapezohedron (albeit with different edge lengths!), covering and volume yield a 2+4+2 shape with 2-fold symmetry, again with different edge lengths.

I did exactly the same investigation. Volume is not working for me (or it is very slow?).
Very often Energy lead to the same result as Covering, sometimes as Packing, but with a better accuracy (more symmetries, less different lengths).
- I agree with your results for 4, 5, 6, 7 and 12.
- For 8 as I said the Covering result is very interesting. But I would call it 2-2-2-2 (not 2-4-2).
- For 9 if you turn your 1-4-4 shape (which I would rather call 1-4-2-4) in another direction, you will find the 3-3-3 again (with different sizes).
- 10 leads to a 2-4-2-2 for Packing (very interesting indeed) and a 1-4-4-1 for the others.
- 11 is hard to understand. Packing leads to a dodecahedron with a missing face (I did a D10 as a dodecahedron with 2 opposite faces missing). And with Energy two pentagons of the original dodecahdron have fused into one elongated hexagon. Covering is unclear...
- 13 leads to 1-4-4-4 (the one I did) for Packing and to something like 1-2-2-4-2-2 (not sure of the order) the the two others. I'd like to make this one...
- 14 is 1-4-2-2-4-1 for Packing (new one) and 1-6-6-1 for the others.
- 15 is very interesting. It always leads to a 3-3-3-3-3 shape but only Coverage has a 3-fold symmetry. The others are enantiomers.

I still have to look at the others, but i can say that this exploration is very exciting.

Hey, I only thought of you the other day. It all seemed to kick off with those D32's I forced you to make for me . I had one re-numbered, (picture to follow) although I am now thinking of downsizing to something like a D14 (curiously).

Although I am unable to donate at this time, I will try to make people aware of the KickStarter Project as I have recently joined the Board Game Designers Forum. It seems that most of the members in the group use dice of some kind.

If you can make people being aware of this project on game forums, I would really appreciate, thanks.
And you are right, before the D32, the only truncated sphere I had done was the D6, which is not really original
So, thank you also for giving me this tricky problem to solve!

Here is a picture of one of the higher scoring dice (D32). A friend of mine with a nice handwriting style was responsible for the numbers. I used small white self adhesive circles (yet to fall off). It is a vast improvement on the ones that I numbered.